13.3: Cognition in Middle Adulthood
-
- Last updated
- Save as PDF
Learning Objectives
By the end of this section, you will be able to:
- Define postformal thinking
- Describe the characteristics of memory and problem-solving in middle adulthood
- Identify changes in intelligence across middle adulthood
- Explain how creativity is expressed in middle adulthood
Charlie was only twenty-two years old when they began graduate studies in science. They spent five years earning a PhD in biology and decades afterward teaching, conducting research, attending conferences, and reading the published work of others. It was not until age fifty years that they won an international prize for their groundbreaking work on the effects of climate change. Charlie’s research has spurred important discussion and debate, and their innovative ideas were founded on years of varied experiences in the classroom, lab, and in the field. Charlie’s intellectual success demonstrates how the journey of adulthood allows us to observe, learn, make connections, and build the wisdom that allows midlife adults to produce important intellectual contributions to science, art, and everyday life.
Overall, cognition seems to reach a pinnacle in early adulthood, but changes in cognition in middle age appear to bring some advantages, including the development of expertise and improved creativity. In this section, you’ll review how changes in adult memory, problem-solving, and intelligence facilitate continued cognitive growth in middle adulthood.
Formal and Postformal Thought
According to Piaget’s theory of cognitive development, the final stage of cognition is formal operational thought , characterized by the capacity for hypothetical, deductive, and abstract thinking. It emerges in adolescence and continues throughout adulthood, though Piaget himself and others have argued that demonstration of some aspects of formal operational thinking is not universal among adults and may be most evident in an adult’s area of expertise (Kuhn, 2008; Piaget, 1972). Other researchers have argued that some adults develop beyond formal operational thinking to a stage known as postformal thought (Perry, 1968). In postformal thought , the intuitive thinking that comes from experience is combined with the logic of formal operations to produce a cognitive style known as relativism. It is recognized in relativism that some problems or dilemmas may have more than one viable solution, depending on a person’s perspective and priorities.
The differences in cognition between formal and postformal thought are subject to debate, and the evidence is unclear as to whether this style of thinking qualifies as a more advanced stage and whether it is typical of adult thinking (Blanchard-Fields, 2001). Whatever the case, changes in cognition that occur in middle adulthood are best understood as changes in the way individuals perceive and process information, rather than as overall advancements or declines. It is possible that the use of relativistic thinking is, in fact, an outcome of the experiences and wisdom accumulated from the lived experience of adults over the lifespan.
Memory and Problem-Solving
Perhaps the most noted research in the understanding of cognition over the course of adulthood is the Seattle Longitudinal Study. The study was initiated by K. Warner Schaie in 1956,, and since that time, it has assessed the psychological and cognitive development of 6,000 adults with ages ranging from 22 to over 100 years old (Seattle Longitudinal Study, 2024). The longitudinal nature of this study provides a great advantage over the previous cross-sectional studies in understanding individual changes in cognitive development because cross-sectional approaches were confounded by generational differences known as cohort effects. The results from the Seattle Longitudinal Study provide insight into changes in adult cognition as people age.
The longitudinal study found that different types of memory and cognitive processes exhibit different trajectories with age. Verbal and semantic memory—long-term recall of words or concepts—remain consistent and even somewhat increase until the end of middle adulthood (Hedden & Gabrieli, 2004; Park et al., 2002; Schaie, 1996). Working memory—the ability to process new and previously encoded information during short-term cognitive tasks—becomes somewhat less efficient as processing speed and short-term storage capacity peak in young adulthood and gradually decline as we age (Park et al., 2002; Rhodes et al., 2019; Schaie, 2003). However, effective memory strategies and engaging in physical exercise can improve working memory ability in middle adulthood (Hertzog et al., 2019; Liu & Lachman, 2019).
Practical problem-solving on everyday tasks, such as purchasing goods or figuring out a bus schedule, shows improvement in middle adulthood, largely from accumulated experience (Chen et al., 2017). Another advantage of middle adulthood is the development of expertise over the course of many years in a career or field of interest. Through continued deliberate practice and engagement, experts in a field can use automatic processing, better strategies, and more creativity in problem-solving for their domain of experience (Ericsson, 2020).
A model of physical and cognitive aging proposed by Nancy Denney suggests that the development of physical abilities and the development of cognitive abilities both follow a similar pattern, with an early adulthood peak and gradual decline (Denney, 1984). Individuals can influence their own peaks by prioritizing the practice of particular skills. Those skills that are not practiced and that therefore peak at lower levels will decline sooner during the aging process. For example, more time spent in formal education results in higher cognitive functions in middle adulthood (Lövdén et al. 2020). As you find in Chapter 15 Physical and Cognitive Development in Late Adulthood (Age 60 and Beyond), diagnosis of cognitive declines is based on comparing individual performance to a universal average, and thus those with higher peaks will take longer to decline to a level at which a diagnosis of cognitive impairment may be made.
It Depends: Brain Games
You may have seen advertisements about games that claim to increase cognitive functions and reduce the chances of having Alzheimer’s disease . These products often include puzzles, riddles, and memory games.
Regular practice of cognitive skills helps advance cognitive abilities and promotes the attainment of higher developmental peaks. Research thus far in cognitive skills training for reduction in cognitive decline is promising but inconclusive. It appears that engaging in games, crafts, and social activity is associated with reduced incidence of cognitive impairment (Krell-Roesch et al., 2017).
Preventing Alzheimer’s with Cognitive Training (PACT) is a clinical trial studying whether there may be a causal link with computerized cognitive training and the reduction of cognitive impairment (Nicholson et al., 2022). In the meantime, the Alzheimer Society of Canada encourages adults to “cross-train your brain” with a variety of activities, such as playing chess or sudoku, learning a new language, and even playing video games to promote cognitive flexibility (Alzheimer Society of Canada, n.d.).
Intelligence and Wisdom
Psychologists have conceptualized intelligence through a variety of theories that are useful in framing how the development of intelligence may vary, depending on how it is defined and measured. As discussed in 7.3 Intelligence in Middle Childhood, Robert Sternberg’s triarchic theory of intelligence proposes that intelligence can be divided into three abilities: analytical, creative, and practical. Sternberg argued that while each of these intellectual components allows people to adapt to changing life demands and circumstances, how they are applied and the priority that they have in our lives will change with age (Berg & Sternberg, 1985). For example, analytical intelligence uses academic knowledge and computation, which can be applied in middle adulthood within a given career or field of interest. Practical intelligence is needed for solving everyday challenges, and as research has shown, adults exhibit improvement in this type of problem-solving as a result of lived experience (Chen et al., 2017). The innovative problem-solving that belies creative intelligence is often built on existing expertise in a given area, and as you have learned, adults continue to increase their semantic memory and expertise throughout middle adulthood.
The value of accumulated knowledge or wisdom in supporting the triarchic domains of intelligence may remind you of Raymond Cattell’s theory of intelligence, also introduced in 7.3 Intelligence in Middle Childhood. Sternberg and colleagues agree and assert that their theory of how intelligence develops in adulthood is certainly compatible with that of Cattell (Berg & Sternberg, 1985). Cattell’s theory of intelligence suggests two primary components of intelligence. Crystalized intelligence consists of accumulated abilities in long-term memory, such as knowledge and vocabulary. Because crystallized intelligence is developed through experience, it exhibits continuous and steady increases throughout the lifespan. Fluid intelligence, which is the ability to reason and solve novel and abstract problems, peaks in early adulthood, remains relatively steady in the middle adult years, and declines as individuals enter older adulthood (Figure 13.11).
There are also generational changes in performance on intelligence tests, known as the Flynn effect . The Flynn effect occurs when newer generations perform better on intelligence tests than previous generations. This doesn’t mean, however, that younger generations are more intelligent than their older counterparts. According to James Flynn, who discovered the phenomenon, explanations include improved access to nutrition and education over time that promote physical and cognitive growth and well-being (Flynn et al., 2012).
Cultural context influences how people think about and apply intellectual abilities (Benson, 2003; Warne, 2020). While Western cultures often perceive intelligence as an analytical skill that helps us to categorize concepts and form arguments, East Asian cultures are more likely to emphasize the interpersonal implications of intelligence and how it helps us to effectively interact with others (Nisbett, 2004; Yang & Sternberg, 1997). 1 Different cultures have also been found to vary in the extent to which they foster and value practical, as opposed to academic, intelligence (Sternberg et al., 2001).
Wisdom is frequently associated with age, and like many cognitive abilities, it develops over time. Like intelligence, conceptions of wisdom may vary cross-culturally, but wisdom is broadly defined by the German psychologist Paul Baltes as “expert knowledge concerning the fundamental pragmatics of life,” a description that incorporates aspects of self-understanding, knowledge, prosocial values, and openness (Bangen et al., 2013). Wisdom can be measured using performance or self-report measurements. Performance measures, such as the Berlin Wisdom Paradigm developed by Baltes and Staudinger (2000), ask respondents to advise fictional characters faced with life decisions: for example, “In reflecting over their life, people sometimes realize that they have not achieved what they had once wanted to achieve. What could a person consider and do in such a situation?” The participant’s level of wisdom is then evaluated based on their response. Self-report measurements, such as the Brief Wisdom Screening Scale (Glück et al., 2013), ask respondents to indicate how much they agree with statements such as, “I can accept the impermanence of things” or “I have grown as a result of the losses I have suffered.”
Research findings about aging and wisdom are complicated. Many aspects of the trait are only marginally predicted by age, if at all (Dong et al., 2023). Across findings, it seems wisdom is less about age and more closely related to applying life experience to difficult life problems or social issues (Figure 13.12). Even personality factors like openness and creativity come into play, though perceptions of the traits associated with wisdom vary cross-culturally. In Western cultures, such as America and Australia, traits such as “experienced” and “knowledgeable” are most closely associated with wisdom. In Eastern cultures, such as India and Japan, wisdom is more closely associated with the social skill of “discretion” (Takahashi & Bordia, 2000). 2
Creativity
While young creative minds may be more popularly visible in some domains, such as music and media, people often reach greater moments of creativity in middle age and beyond. A study of famous painters from the nineteenth and twentieth centuries found that these artists typically created their most valuable paintings when they were forty-two years old (Franses, 2013). Creativity is often thought of as only an artistic quality, but it is derived from cognitive abilities that can be applied in several domains, including science. In fact, an examination of scientists who won the Nobel Prize for scientific breakthroughs found that since the year 2000, most prize-winning physicists and chemists were well into middle adulthood when they made their discoveries, with physicists being forty-eight years old on average (Jones & Weinberg, 2011). As we age, we learn what works best for each of our own creative processes, allowing us to maximize individual creativity.
Creativity can be generated by both divergent and convergent thinking. Divergent thinking allows us to develop many solutions to a problem when there is no clear solution (Guilford, 1967). This can mean applying the abilities of fluency, flexibility, originality, and elaboration. If you had to come up with all the things you might do with a traditional red brick, for example, you produce many ideas (fluency), a wide variety of ideas (flexibility), unusual ideas (originality), and add complex details to your ideas (elaboration) (Figure 13.13). Convergent thinking is the creative ability to determine a best solution to produce a desired outcome. In the tale of the “Three Little Pigs” who each construct a house to keep safe from the big bad wolf, the pig who chooses to build his house from bricks (and not straw or sticks) demonstrates convergent thinking.
Link to Learning
Many historical individuals like Robert Frost, Virginia Woolf, and Charles Darwin have produced their best work around the age of forty years and continued into their fifties and beyond (Jones & Weinberg, 2011; Simonton, 2000). Check out this list of great minds compiled by the American Alliance of Museums to read about how people often continue to produce and earn recognition well into late adulthood.
References
Alzheimer Society of Canada. (n.d.). Challenging your brain. Alzheimer Society of Canada. https://alzheimer.ca/en/help-support...ing-your-brain
Baltes, P. B., & Staudinger, U. M. (2000). Wisdom: A metaheuristic (pragmatic) to orchestrate mind and virtue toward excellence. American Psychologist, 55 (1), 122–136. doi.org/10.1037/0003-066X.55.1.122
Bangen, K. J., Meeks, T. W., & Jeste, D. V. (2013). Defining and assessing wisdom: A review of the literature. The American Journal of Geriatric Psychiatry, 21 (12), 1254–1266. https://doi.org/10.1016/j.jagp.2012.11.020
Benson, E. (2003). Intelligence across cultures. American Psychological Association Monitor, 34 (2), 56. https://www.apa.org/monitor/feb03/intelligence
Berg, C. A., & Sternberg, R. J. (1985). A triarchic theory of intellectual development during adulthood. Developmental Review, 5(4), 334–370. https://doi.org/10.1016/0273-2297(85)90017-6
Blanchard-Fields, F. (2001). Adult cognitive development: Post-Piagetian perspectives. In N. J. Smelser & P. B. Baltes (Eds.), International encyclopedia of the social & behavioral sciences (pp. 132–135). Pergamon. https://doi.org/10.1016/B0-08-043076-7/01564-3
Chen, X., Hertzog, C., & Park, D. C. (2017). Cognitive predictors of everyday problem solving across the lifespan. Gerontology , 63 (4), 372–384. doi.org/10.1159/000459622
Denney, N. W. (1984). A model of cognitive development across the life span. Developmental Review , 4 (2), 171–191. https://doi.org/10.1016/0273-2297(84)90006-6
Dong, M., Weststrate, N. M., & Fournier, M. A. (2023). Thirty years of psychological wisdom research: What we know about the correlates of an ancient concept. Perspectives on Psychological Science: A Journal of the Association for Psychological Science , 18 (4), 778–811. doi.org/10.1177/17456916221114096
Ericsson, K. A. (2020). Towards a science of the acquisition of expert performance in sports: Clarifying the differences between deliberate practice and other types of practice. Journal of Sports Sciences , 38 (2), 159–176. doi.org/10.1080/02640414.2019.1688618
Flynn, J., Shaughnessy, M. F., & Fulgham, S. W. (2012). Interview with Jim Flynn about the Flynn effect. North American Journal of Psychology, 14 (1), 25–38.
Franses, P. H. (2013). When do painters make their best work? Creativity Research Journal, 25 (4), 457–462, doi.org/10.1080/10400419.2013.843912
Glück, J., König, S., Naschenweng, K., et al. (2013). How to measure wisdom: Content, reliability, and validity of five measures. Frontiers in Psychology , 4 . https://doi.org/10.3389/fpsyg.2013.00405
Guilford, J. P. (1967). The nature of human intelligence. McGraw Hill.
Hedden, T., & Gabrieli, J. D. E. (2004). Insights into the ageing mind: A view from cognitive neuroscience. Nature Reviews Neuroscience, 5 (2), 87–96. https://doi.org/10.1038/nrn1323
Hertzog, C., Lustig, E., Pearman, A., & Waris, A. (2019). Behaviors and strategies supporting everyday memory in older adults. Gerontology , 65 (4), 419–429. doi.org/10.1159/000495910
Jones, B. F., & Weinberg, B. A. (2011). Age dynamics in scientific creativity. PNAS, 108 (47), 18910–18914, doi.org/10.1073/pnas.1102895108
Krell-Roesch, J., Vemuri, P., Pink, A., Roberts, R. O., Stokin, G. B., Mielke, M. M., Christianson, T. J. H., Knopman, D. S., Petersen, R. C., Kremers, W. K., & Geda, Y. E. (2017). Association between mentally stimulating activities in late life and the outcome of incident mild cognitive impairment, with an analysis of the APOE ɛ4 genotype. JAMA Neurology , 74 (3), 332–338. doi.org/10.1001/jamaneurol.2016.3822
Kuhn, D. (2008). Formal operations from a twenty-first century perspective. Human Development, 51 (1), 48–55. doi.org/10.1159/000113155
Liu, Y., & Lachman, M. E. (2019). Socioeconomic status and parenting style from childhood: Long-term effects on cognitive function in middle and later adulthood. The Journals of Gerontology: Series B , 74 (6), e13–e24. doi.org/10.1093/geronb/gbz034
Lövdén, M., Fratiglioni, L., Glymour, M. M., Lindenberger, U., & Tucker-Drob, E. M. (2020). Education and cognitive functioning across the life span. Psychological Science in the Public Interest : A Journal of the American Psychological Society , 21 (1), 6–41. doi.org/10.1177/1529100620920576
Nicholson, J. S., Hudak, E. M., Phillips, C. B., Chanti-Ketterl, M., O’Brien, J. L., Ross, L. A., Lister, J. L., Burke, J. R., Potter, G., Plassman, B. L., Woods, A. J., Krischner, J., & Edwards, J. D. (2022). The preventing Alzheimer’s with cognitive training (PACT) randomized clinical trial. Contemporary clinical trials , 123 , Article 106978. https://doi.org/10.1016/j.cct.2022.106978
Nisbett, R. E. (2004). The geography of thought . Free Press.
Park, D. C., Lautenschlager, G., Hedden, T., Smith, A.D., & Smith, P. K. (2002). Models of visuospatial and verbal memory across the adult life span. Psychology and Aging, 17 (2), 299–320. doi.org/10.1037/0882-79<IP_ADDRESS>9
Paulhus, D., Wehr, P., Harms, P., & Strasser, D. (2002). Use of exemplar surveys to reveal implicit types of intelligence. Personality and Social Psychology Bulletin, 28 , 1051–1062. https://doi.org/ 10.1177/01461672022811004
Perry, W. B. (1968). Forms of intellectual and ethical development in the college years: A scheme. Hold, Rinehart, and Winston.
Piaget, J. (1972). Intellectual evolution from adolescence to adulthood. Human Development, 15 (1), 1–12. doi.org/10.1159/000271225
Rhodes, S., Jaroslawska, A. J., Doherty, J. M., Naveh-Benjamin, M., Cowan, N., Camos, V., Barrouillet, P., & Logie, R. H. (2019). Storage and processing in working memory: Assessing dual-task performance and task prioritization across the adult lifespan. Journal of Experimental Psychology: General , 148 (7), 1204. doi.org/10.1037/xge0000539
Schaie, K. W. (1996). Intellectual development in adulthood: The Seattle longitudinal study. Cambridge University Press.
Schaie, K.W. (2003, January 23–24). Cognitive aging. National Research Council. https://sls.psychiatry.uw.edu/wp-con...-aging2003.pdf
Schaie K. W., & Willis, S. L. (2010). The Seattle longitudinal study of adult cognitive development. ISSBD Bull. 2010; 57 (1), 24–29. www.ncbi.nlm.nih.gov/pmc/articles/PMC3607395/
Seattle Longitudinal Study. (2024). Seattle longitudinal study . Department of Psychiatry and Behavioral Sciences, UW Medicine. https://sls.psychiatry.uw.edu/
Simonton, D. K. (2000). Creativity: Cognitive, personal, developmental, and social aspects. American Psychologist, 55 (1), 151–158. doi.org/10.1037/0003-066X.55.1.151
Sternberg, R. J., Nokes, C., Geissler, P. W., Prince, R., Okatcha, F., Bundy, D. A., & Grigorenko, E. L. (2001). The relationship between academic and practical intelligence: A case study in Kenya. Intelligence, 29 (5), 401-418. https://doi.org/10.1016/S0160-2896(01)00065-4
Takahashi, M., & Bordia, P. (2000). The concept of wisdom: A cross-cultural comparison. International Journal of Psychology, 35 (1). https://doi.org/ 10.1080/002075900399475
Warne, R. T. (2020). Intelligence is a Western concept that does not apply to non-Western cultures. In In the know: Debunking 35 myths about human intelligence (pp. 46–51). Cambridge University Press.
Yang, S.-Y., & Sternberg, R. J. (1997). Conceptions of intelligence in ancient Chinese philosophy. Journal of Theoretical and Philosophical Psychology, 17 (2), 101–119. doi.org/10.1037/h0091164
Footnotes
- 1 The study (Nisbett, 2004) uses the terms “East Asian” and “Western.” The study (Yang & Sternberg, 1997) uses the terms “Western” and “Eastern.”
- 2 This study (Takahashi & Bordia, 2000) uses the terms "Western" to include "American" and "Australian" and uses "Eastern" to include "Indian" and "Japanese."


1.1: What is This Book All About?
This book is intended to be used for a one-semester/quarter introduction to proof course (sometimes referred to as a transition to proof course). The purpose of this book is to introduce the reader to the process of constructing and writing formal and rigorous mathematical proofs. The intended audience is mathematics majors and minors. However, this book is also appropriate for anyone curious about mathematics and writing proofs. Most users of this book will have taken at least one semester of calculus, although other than some familiarity with a few standard functions in Chapter 8, content knowledge of calculus is not required. The book includes more content than one can expect to cover in a single semester/quarter. This allows the instructor/reader to pick and choose the sections that suit their needs and desires. Each chapter takes a focused approach to the included topics, but also includes many gentle exercises aimed at developing intuition.
The following sections form the core of the book and are likely the sections that an instructor would focus on in a one-semester introduction to proof course.
- Chapter 2: Mathematics and Logic. All sections.
- Chapter 3: Set Theory. Sections 3.1, 3.3, 3.4, and 3.5.
- Chapter 4: Induction. All sections.
- Chapter 7: Relations and Partitions. Sections 7.1, 7.2, and 7.3.
- Chapter 8: Functions. Sections 8.1, 8.2, 8.3, and 8.4.
- Chapter 9: Cardinality. All sections.
Time permitting, instructors can pick and choose topics from the remaining sections. I typically cover the core sections listed above together with Chapter 6: Three Famous Theorems during a single semester. The Instructor Guide contains examples of a few possible paths through the material, as well as information about which sections and theorems depend on material earlier in the book.


6.1: The Wave Equation
A wave can be described by a function \(f(x,t)\) , called a wavefunction , which specifies the value of a measurable physical quantity at each position \(x\) and time \(t\) . For simplicity, we will assume that space is one-dimensional, so \(x\) is a single real number. We will also assume that \(f(x,t)\) is a number, rather than a more complicated object such as a vector. For instance, a sound wave can be described by a wavefunction \(f(x,t)\) representing the air pressure at each point of space and time.
The evolution of the wavefunction is described by a partial differential equation (PDE) called the time-dependent wave equation : \[\frac{\partial^2 f}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}, \;\;\; v \in\mathbb{R}^+.\] The parameter \(v\) , which we take to be a positive real constant, is called the wave speed , for reasons that will shortly become clear.
Sometimes, we re-arrange the wave equation into the following form, consisting of a linear differential operator acting on \(f(x,t)\) : \[\left(\frac{\partial^2}{\partial x^2} - \frac{1}{v^2} \frac{\partial^2}{\partial t^2}\right) \; f(x,t) = 0.\] This way of writing the wave equation emphasizes that it is a linear PDE, meaning that any linear superposition of solutions is likewise a solution.


7.2: Transformation of Vectors
Learning Objectives
- Explain how to transform vectors
Now suppose we want to transform a vector whose components are expressed in the \((T,X)\) coordinates into components expressed in \((t,x)\).
Our most basic example of a vector is a displacement \((∆T,∆X)\), and if we make this an infinitesimal \((dT,dX)\) then we don’t need to worry about the fact that the chart in figure \(\PageIndex{1}\) has curves on it — close up, curves look like straight lines. 1 If we think of the coordinate \(t\) as a function of two variables, \(t = t(T,X)\), then \(t\) is changing for two different reasons: its first input \(T\) changes, and also its second input \(X\). If \(t\) were only a function of one variable \(t(T)\), then the change in \(t\) would be given simply by the chain rule, \(dT = \frac{dt/dT}{d} T\). Since it actually has two such reasons to change, we add the two changes:
\[dt = \frac{\partial t}{\partial T} dT + \frac{\partial t}{\partial X} dX\]
The derivatives are partial derivatives, and these derivatives exist because, as we will always assume, the change of coordinates is smooth. An exactly analogous expression applies for \(dx\).
\[dx = \frac{\partial x}{\partial T} dT + \frac{\partial x}{\partial X} dX\]
Before we carry out the details of this calculation, let’s stop and note that the results so far are completely general. Since we have so far made no use of the actual equations for this particular change of coordinates, these expressions would apply to any such transformation, including the special cases we’ve encountered so far, such as Lorentz transformations and scaling. (For example, if we’d been scaling by a factor \(α\), then all of the partial derivatives would simply have equaled \(α\).). Furthermore, our definition of a vector is that a vector is anything that transforms like a vector. Since we’ve established that the rules above apply to a displacement vector, we conclude that they would also apply to any other vector, say an energy-momentum vector.
Returning to this specific example, application of the facts
\[\frac{\mathrm{d} \sinh u}{\mathrm{d} u} = \cosh u\; \; \text{and}\; \; \frac{\mathrm{d} \cosh u}{\mathrm{d} u} = \sinh u\]
tells us that the vector \((dT,dX)\) is transformed to:
\[(dt,dx) = (X \cosh T dT + \sinh T dX ,\; X \sinh T dT + \cosh T dX)\]
As an example of how this applies universally to any type of vector, suppose that the observer aboard a spaceship with world-line \((T,X) = (τ,1)\) has a favorite paperweight with mass \(m\). According to measurements carried out aboard her ship, its energy-momentum vector is:
\[(p_T,p_X) = (m,0)\]
In the unaccelerated coordinates, this becomes
\[\begin{align*} (p_t,p_x) &= (X \cosh T p_T + \sinh T p_X ,\; X \sinh T p_T + \cosh T p_X)\\ &= (mX\cosh T,\; mX\sinh T)\\ &= (m\cosh \tau ,\; m\sinh \tau ) \end{align*}\]
Since the functions cosh and sinh behave like \(e^x\) for large \(x\), we find that after the astronaut has spent a reasonable amount of proper time \(τ\) accelerating, the paperweight’s mass-energy and momentum will have grown to the point where it’s an awesome weapon of mass destruction, capable of obliterating an entire galaxy.
References
1 Here we make use of the fact that the change of coordinate was smooth, i.e., a diffeomorphism. Otherwise the curves could have kinks in them that would still look like kinks under any magnification.


3.6: Stress
A stressor is anything that causes stress. Even positive life events such as the birth of a baby or a wedding can cause people stress. Life events such as weddings, divorce, death, births of children, and retirement can cause people stress. Additional stressors affecting the elderly may include financial problems, death of a spouse, family, or close friends, physical changes, pain, and loss of independence.
When we feel stress, adrenaline is released by the endocrine system. It causes our blood pressure, heart rate, and respiratory rate to increase. Stress that is not relieved can result in long-term physical problems such as high blood pressure. It can also cause anxiety, depression, aggression, and mood swings. Some people manage stress by engaging in unhealthy behaviors such as overeating, smoking, yelling at those near them, or drinking alcohol.
If Home Health Aides/Personal Care Aides observe unhealthy coping behaviors from their patient and high levels of stress, they should discuss their concerns with their supervisor and work with him or her to develop healthy coping strategies for the patient.
1. Which of the following events can cause stress to a person?
- a). New marriage
- b). Birth of a baby
- c). Death of a spouse
- d). Retirement
- e). Loss of income
- f). New diagnosis of heart disease
- g). Pain
- h). Losing the ability to dress oneself
- Answer
-
All of these events can cause a person to feel stress.
Feedback :
Both positive and negative stressors can contribute to a feeling of stress and cause a person anxiety, insomnia, high blood pressure, and other health problems. Stressors include a new marriage or one that has problems, the birth of a baby, the death of a spouse or loved one, retirement, loss of income, a new diagnosis of any disease or illness, pain, and losing one’s independence and ability to perform tasks independently.
How to Help a Patient Effectively Manage Stress
- Encourage and provide healthy, nutritious food
- Encourage and work with the patient on maintaining physical activity or an exercise plan
- Promote good sleep by providinga peaceful sleeping environment
- Find activities the patient enjoys such as reading, listening to music, watching movies, or spending time with friends
- Teach the patient relaxation techniques such as deep breathing
- Provide a warm bath and back massage for the patient
- Apply lotion to the patient’s legs and hands
- If the patient wishes, pray with them or organize a clergy visit
- Allow the patient space to be alone for short periods to relax and think
- Provide space and time for the patient to talk and encourage them to express their thoughts and feelings
Which of the following are healthy ways to manage stress?
a). Smoking
b). Overeating
c). Talking to a trusted friend
d). Drinking alcohol
e). Taking a warm bath
- Answer
-
1. C and E
Feedback :
1. Healthy ways to manage stress include talking to a trusted friend, taking a warm bath, eating nutritious food, engaging in regular exercise, praying, and taking time to relax. Smoking, overeating, and drinking alcohol excessively contribute to stress.


1.E: Kinematics (Exercise)
Conceptual Questions
2.1: Displacement
1. Give an example in which there are clear distinctions among distance traveled, displacement, and magnitude of displacement. Specifically identify each quantity in your example.
2. Under what circumstances does distance traveled equal magnitude of displacement? What is the only case in which magnitude of displacement and displacement are exactly the same?
3. Bacteria move back and forth by using their flagella (structures that look like little tails). Speeds of up to \(50 \ \mu \mathrm{m} / \mathrm{s} \ \left(50 \times 10^{-6} \mathrm{~m} / \mathrm{s}\right)\) have been observed. The total distance traveled by a bacterium is large for its size, while its displacement is small. Why is this?
2.2: Vectors, Scalars, and Coordinate Systems
4. A student writes, “ A bird that is diving for prey has a speed of −10 m/s.” What is wrong with the student’s statement? What has the student actually described? Explain.
5. What is the speed of the bird in Exercise 2.2.4 ?
6. Acceleration is the change in velocity over time. Given this information, is acceleration a vector or a scalar quantity? Explain.
7. A weather forecast states that the temperature is predicted to be −5ºC the following day. Is this temperature a vector or a scalar quantity? Explain.
2.3: Time, Velocity, and Speed
8. Give an example (but not one from the text) of a device used to measure time and identify what change in that device indicates a change in time.
9. There is a distinction between average speed and the magnitude of average velocity. Give an example that illustrates the difference between these two quantities.
10. Does a car’s odometer measure position or displacement? Does its speedometer measure speed or velocity?
11. If you divide the total distance traveled on a car trip (as determined by the odometer) by the time for the trip, are you calculating the average speed or the magnitude of the average velocity? Under what circumstances are these two quantities the same?
12. How are instantaneous velocity and instantaneous speed related to one another? How do they differ?
2.4: Acceleration
13. Is it possible for speed to be constant while acceleration is not zero? Give an example of such a situation.
14. Is it possible for velocity to be constant while acceleration is not zero? Explain.
15. Give an example in which velocity is zero yet acceleration is not.
16. If a subway train is moving to the left (has a negative velocity) and then comes to a stop, what is the direction of its acceleration? Is the acceleration positive or negative?
17. Plus and minus signs are used in one-dimensional motion to indicate direction. What is the sign of an acceleration that reduces the magnitude of a negative velocity? Of a positive velocity?
2.5: Motion Equations for Constant Acceleration in One Dimension
18. In a constant-acceleration motion, how does the position change? How does the velocity change?
19. Describe the relationship between the position and the velocity in a constant-acceleration motion.
20. Describe the relationship between the velocity and the acceleration in a constant-acceleration motion.
21. In a constant-acceleration motion starting from rest, how does the distance traveled change with the duration of motion? If the motion continues for double the time (total), how much does the distance traveled increase (total)?
2.6: Falling Objects
22. What is the acceleration of a rock thrown straight upward on the way up? At the top of its flight? On the way down?
23. An object that is thrown straight up falls back to Earth. This is one-dimensional motion. (a) When is its velocity zero? (b) Does its velocity change direction? (c) Does the acceleration due to gravity have the same sign on the way up as on the way down?
24. Suppose you throw a rock nearly straight up at a coconut in a palm tree, and the rock misses on the way up but hits the coconut on the way down. Neglecting air resistance, how does the speed of the rock when it hits the coconut on the way down compare with what it would have been if it had hit the coconut on the way up? Is it more likely to dislodge the coconut on the way up or down? Explain.
25. If an object is thrown straight up and air resistance is negligible, then its speed when it returns to the starting point is the same as when it was released. If air resistance were not negligible, how would its speed upon return compare with its initial speed? How would the maximum height to which it rises be affected?
26. The severity of a fall depends on your speed when you strike the ground. All factors but the acceleration due to gravity being the same, how many times higher could a safe fall on the Moon be than on Earth (gravitational acceleration on the Moon is about 1/6 that of the Earth)?
27. How many times higher could an astronaut jump on the Moon than on Earth if his takeoff speed is the same in both locations (gravitational acceleration on the Moon is about 1/6 of \(g\) on Earth)?
2.7: Projectile Motion
28. Answer the following questions for projectile motion on level ground assuming negligible air resistance (the initial angle being neither \(0^{\circ}\) nor \(90^{\circ}\)): (a) Is the velocity ever zero? (b) When is the velocity a minimum? A maximum? (c) Can the velocity ever be the same as the initial velocity at a time other than at \(t=0\)? (d) Can the speed ever be the same as the initial speed at a time other than at \(t=0\)?
29. Answer the following questions for projectile motion on level ground assuming negligible air resistance (the initial angle being neither \(0^{\circ}\) nor \(90^{\circ}\)): (a) Is the acceleration ever zero? (b) Is the acceleration ever in the same direction as a component of velocity? (c) Is the acceleration ever opposite in direction to a component of velocity?
30. For a fixed initial speed, the range of a projectile is determined by the angle at which it is fired. For all but the maximum, there are two angles that give the same range. Considering factors that might affect the ability of an archer to hit a target, such as wind, explain why the smaller angle (closer to the horizontal) is preferable. When would it be necessary for the archer to use the larger angle? Why does the punter in a football game use the higher trajectory?
31. During a lecture demonstration, a professor places two coins on the edge of a table. She then flicks one of the coins horizontally off the table, simultaneously nudging the other over the edge. Describe the subsequent motion of the two coins, in particular discussing whether they hit the floor at the same time.
2.8: Centripetal Acceleration
32. Give examples of centripetal acceleration from everyday experience.
33. A satellite in a circular orbit around the Earth is undergoing a centripetal acceleration. Explain how this description is consistent with the description that the satellite in free-fall.
34. Can centripetal acceleration change the speed of circular motion? Explain.
Problems & Exercises
2.1: Displacement
35.
Find the following for path A in Figure \(\PageIndex{1}\): (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
Answer
(a) 7 m
(b) 7 m
(c) +7 m
36. Find the following for path B in Figure \(\PageIndex{1}\): (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
37. Find the following for path C in Figure \(\PageIndex{1}\): (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
Answer
(a) 13 m
(b) 9 m
(c) +9 m
38. Find the following for path D in Figure \(\PageIndex{1}\): (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.


1.2: Activities
Things You Will Need
All you will need for this lab is the PhET simulator , which you can run now in a separate window.
Configuring the Simulator
Begin with the "Energy" panel of the simulator. Configure it as follows:
- Open the "Particles" and "Injection Temperature" dropdowns (the "Average Speed" and "Speed" dropdowns should already be open, but if they are not, then open them as well).
- Select the units for the Pressure gauge to be kilopascals (kPa).
- Uncheck the "Collisions" box in the "Particles" dropdown.
A few useful features of the simulator to know about:
- All the particles injected into the chamber come in with the same kinetic energy (which is proportional to their "temperature"). The injection temperature can be selected, or by default, the injection temperature is the same as the current temperature of the gas in the chamber.
- You can inject 50 particles at a time of a single variety with the button (fully extending the pump does the same, if you are into the animation). Using the button injects one particle at a time. The similar buttons whose arrows point to the left remove particles in the same number. [These removal buttons are of little use, as they choose particles at random to remove, and may select more "hot" particles than "cold" ones, meaning that while we can add particles without changing the gas temperature, we cannot remove them without changing the gas temperature.]
- You can remove all of the particles from the chamber at once with the button.
Deducing the Simulated Gases
We will start by assuming that this simulator is designed to demonstrate the three-dimensional kinetic theory of gases. Two gases are selected (one that is "heavy," and the other that is "light"), and in an effort to be as accurate as possible, the quantitative read-outs of the simulator are based on the correct masses of these gases.
- Your task is to determine which of the five gases listed below is represented by the light particles, and which one is represented by the heavy particles. You are of course free to look up properties of these atoms and molecules to help you make this determination.
\[\text{H}_2\;,\;\;\;\text{He}\;,\;\;\;\text{N}_2\;,\;\;\;\text{O}_2\;,\;\;\;\text{Ne}\nonumber\]
Timeout for Some Math
We now turn toward the question raised in the Background Material about whether this simulator actually does represent a gas in three dimensions. The Maxwell-Boltzmann distribution function gives us a measure of the probability of finding a random particle that has a speed in a given infinitesimal range:
\[P\left(v\leftrightarrow v+dv\right) = \mathcal D\left(v\right)dv\]
This Maxwell-Boltzmann distribution \(\mathcal D\left(v\right)\) is derived using the assumption that the particles can "randomly" change their individual velocities, subject to the constraint that the sum of their kinetic energies remains fixed. In other words, all collisions (with the walls of the container, and with each other) must be elastic. Our simulator is programmed to exactly follow this constraint. Something else that is hidden within the distribution function is the dimension of the space within which the particles are constrained. What this means is that if you select a random particle from a gas, the probability that it will have a speed between \(v\) and \(v+dv\) is different if that gas is constrained to two dimensions than if it occupies three dimensions. This is evidenced by the two distribution functions:
\[\begin{array}{l} \mathcal D_{2d}\left(v\right)=\alpha \;v \;e^{-\frac{1}{2}\alpha v^2} \\ \mathcal D_{3d}\left(v\right)=\sqrt{\frac{2}{\pi}}\;\alpha^{\frac{3}{2}} \;v^2\; e^{-\frac{1}{2}\alpha v^2} \end{array}\]
In both cases, the constant \(\alpha\) is defined in terms of the temperature of the gas and the mass of the particles (and the Boltzmann constant \(k_B\)):
\[\alpha\equiv \dfrac{m}{k_BT}\]
Multiplying each of these by \(dv\) gives the probability of selecting a random particle with a speed that falls in the range \(v\leftrightarrow v+dv\) for its particular dimension. For what follows below, the following definite gaussian integrals will be useful:
\[\begin{array}{l} \int\limits_0^{\infty} x\;e^{-\beta x^2}\;dx && = && \frac{1}{2}\beta^{-\frac{2}{2}} \\ \int\limits_0^{\infty} x^2\;e^{-\beta x^2}\;dx && = && \frac{1}{4}\sqrt\pi\;\beta^{-\frac{3}{2}} \\ \int\limits_0^{\infty} x^3\;e^{-\beta x^2}\;dx && = && \frac{1}{2}\beta^{-\frac{4}{2}} \\ \int\limits_0^{\infty} x^4\;e^{-\beta x^2}\;dx && = && \frac{3}{8}\sqrt\pi\;\beta^{-\frac{5}{2}} \end{array}\]
- The probability of finding a random particle with any speed whatsoever (from \(v=0\) to \(v=\infty\)) must obviously equal 1, and this is the sum of the probabilities of all the tiny ranges (which is of course an integral). Show that each of the two distributions given satisfies this important requirement.
The value of having a probability distribution is that it can be used to compute averages. An average of several possible values that can arise randomly is the product of each value and it corresponding probability, summed over all of the possible values:
\[\text{average of }x=\left<x\right> = x_1\cdot P\left(x_1\right) + x_2\cdot P\left(x_2\right) + \dots\]
When the values lie on a continuum (like particle speeds), this sum becomes an integral. So multiplying every possible speed \(v\) by its corresponding probability \(\mathcal D\left(v\right)dv\), and adding all of these products together gives a means for computing the average speed:
\[\left<v\right> =\int\limits_0^\infty v\;\mathcal D\left(v\right)dv\]
We can similarly take averages of other quantities, most notably the square of the velocity:
\[\left<v^2\right> =\int\limits_0^\infty v^2\;\mathcal D\left(v\right)dv\]
- Determine the relationship (i.e. the ratio) between the average and rms particle speeds for both a two and three-dimensional gas.
Simulated Evaluation
We are at last ready to examine the simulator. The average speed comes as a direct read-out in the simulator. The rms speed we can determine using the trick indicated in the Background Material – if we start with the "no collisions" configuration given above, the particle speeds remain constant and uniform. Then, without changing anything else about the gas, we can turn on collisions to introduce randomness.
-
Determine whether the simulator replicates a two or three-dimensional gas. This consists of several parts:
- Run the simulator "without randomness" to get the particle rms speed.
- Introduce randomness and get the average speed. Does the rms value change when you do this? Why or why not?
- The average speed fluctuates with time, so you will need to "sample" these numbers at random times, say by closing your eyes, waiting a second, and clicking the pause button, then writing down the number. Then un-pause the simulator and repeat several times. An average of these samples should be a good measure of the "true" average speed.
- The data obtained from the sampling method just described also allows you to calculate an uncertainty interval for the measurement. For a refresher on how to compute standard deviation, review this from your very first 9HA lab.
- Determine whether the relationship between the average and rms speeds is (to within uncertainty) characteristic of a two or three-dimensional gas.


14.3.7: Out-of-Pocket Maximum
Most plans also have an annual out-of-pocket maximum, which is the total amount the beneficiary would have to pay that year. Deductibles, copays, and coinsurance all count toward this maximum. So let’s add to our previous example that the person who broke their arm had a $7000 out of pocket maximum. They had already paid their $3500 deductible, then would be responsible for 20% of the remaining costs (coinsurance), up until that $7000 out-of-pocket maximum is met - then the insurance company is responsible for 100% of the remainder of the bill. So let’s say that after this broken arm/hand surgery scenario, they required physical therapy visits. Normally they would have to pay a $50 copayment for each visit, but since their out-of-pocket maximum had been met the insurance company covers the copayment for the rest of the year. Now, if the broken arm had only required the ER visit and an X-ray, and the total bill was $5000, the insured would have to pay the $3500 deductible, then 20% of the remaining $1500 ($300), and would have to pay those $50 copays for physical therapy until the out-of-pocket maximum had been met for the year (Cigna Healthcare, 2023).


11.10: References
Adams, C. (2023, January 17). Jackson, Mississippi, still dealing with water crisis. NBC News . https://www.nbcnews.com/news/nbcblk/...isis-rcna65563
Andrews, H. M., & Rose, M. A. (2018). Understanding pesticides in organic and conventional crop production systems . Ohioline. https://ohioline.osu.edu/factsheet/anr-69
ATSDR. (2024, June 21). Agency for Toxic Substances and Disease Registry . https://www.atsdr.cdc.gov/index.html
Breslow, J. (2022, August 31). The water crisis in Jackson follows years of failure to fix an aging system. NPR . https://www.npr.org/2022/08/31/11201...i-water-crisis
Burke, L. D., & Weill, B. (2023). Public health and society: Current issues . Jones & Bartlett Learning.
California Climate Adaptation Strategy. (2024). Summary of Projected Climate Change Impacts on California . California Climate Adaptation Strategy. https://climateresilience.ca.gov/ove...w/impacts.html
CalRecycle. (2024). New statewide mandatory organic waste collection . CalRecycle Home Page. https://calrecycle.ca.gov/organics/slcp/collection/
CARB. (2022, August 25). California moves to accelerate to 100% new zero-emission vehicle sales by 2035 . California Air Resources Board (CARB). https://ww2.arb.ca.gov/news/californ...cle-sales-2035
Chea, T. (2021, August 17). Climate change in california is threatening the world’s top almond producer. Associated Press (AP) . https://www.npr.org/2021/08/17/10284...ond-production
Climate change impacts in the United States, highlights: U.S. national climate assessment . (2014).
Contributors to Wikimedia projects. (2024, June 11). Jackson, Mississippi . Wikipedia. https://en.Wikipedia.org/wiki/Jackson,_Mississippi
Dialesandro, J., Brazil, N., Wheeler, S., & Abunnasr, Y. (2021). Dimensions of thermal inequity: Neighborhood social demographics and urban heat in the southwestern U.S. International Journal of Environmental Research and Public Health , 18 (3), 941. https://doi.org/10.3390/ijerph18030941
Dixon, J. (n.d.). Time line: How Flint’s water crisis unfolded . Detroit Free Press. Retrieved June 26, 2024, from https://www.freep.com/pages/interact...isis-timeline/
EPA. (1972, December 31). DDT ban takes effect . US EPA; EPA Web Archive. https://www.epa.gov/archive/epa/abou...es-effect.html
EPA. (2013a, March 5). Learn about asbestos . US EPA. https://www.epa.gov/asbestos/learn-about-asbestos
EPA. (2013b, June 4). Water enforcement . US Environmental Protection Agency. https://www.epa.gov/enforcement/water-enforcement
EPA. (2014, August 14). Introduction to indoor air quality . US Environmental Protection Agency. https://www.epa.gov/indoor-air-quali...or-air-quality
EPA. (2015, August 11). Sustainable management of food basics . US EPA. https://www.epa.gov/sustainable-mana...basics#sources
EPA. (2016, February 2). Basic information about lead in drinking water . US EPA. https://www.epa.gov/ground-water-and...drinking-water
EPA. (2018, November 26). Waste management . US EPA. https://www.epa.gov/trinationalanaly...ste-management
EPA. (2023a, November 22). National overview: Facts and figures on materials, wastes and recycling . US EPA. https://www.epa.gov/facts-and-figure...ures-materials
EPA. (2023b, December 8). Information on ozone and ozone depletion . US EPA; U.S. Environmental Protection Agency (EPA). https://www.epa.gov/ozone-layer-prot...zone-depletion
EPA. (2024, June 6). Superfund: National priorities list (NPL) . US EPA. https://www.epa.gov/superfund/superf...ities-list-npl
Evers, J. (2024, April 10). Great Pacific Garbage Patch . National Geographic Society. https://education.nationalgeographic...garbage-patch/
Extreme heat in North America, Europe and China in July 2023 made much more likely by climate change – World Weather Attribution . (n.d.). https://www.worldweatherattribution....limate-change/
Frumkin, H. (2016). Environmental health: From global to local . John Wiley & Sons.
Fry, H., & Reyes-Velarde, A. (2019, February 20). California wastes most of its rainwater, which simply goes down the drain. Los Angeles Times . https://www.latimes.com/local/lanow/...220-story.html
Harris, W. (2023, September 8). How long does it take for plastic to decompose? HowStuffWorks . https://science.howstuffworks.com/sc...biodegrade.htm
Kennedy, M. (2016, April 20). Lead-Laced water in Flint: A step-by-step look at the makings of A crisis. NPR . https://www.npr.org/sections/thetwo-...gs-of-a-crisis
Muhammad, S., & Judin, N. (2024, May 10). EPA finds Mississippi didn’t racially discriminate against Jackson in water funds. Mississippi Free Press . https://www.mississippifreepress.org...ructure-funds/
Murembya, L., & Guthrie, E. (2016). Demographic and Labor Market Profile: City of Flint . State of Michigan Department of Technology, Management, and Budget Bureau of Labor Market Information and Strategic Initiatives. https://milmi.org/_docs/publications...kt_Profile.pdf
NASA. (2024). Evidence . NASA Science; The National Aeronautics and Space Administration. https://science.nasa.gov/climate-change/evidence/
National climate assessment . (n.d.). National Climate Assessment. Retrieved June 19, 2024, from https://nca2014.globalchange.gov/rep...s/human-health
SPLC. (2023). New Orleans homeowners still in financial storm 18 years after Katrina. Southern Poverty Law Center. https://www.splcenter.org/news/2023/...ricane-katrina
NOAA. (2023, August 18). Extreme ocean temperatures are affecting Florida’s Coral Reef . National Environmental Satellite, Data, and Information Service; National Environmental Satellite, Data, and Information Service. https://www.nesdis.noaa.gov/news/ext...das-coral-reef
Oh, L. (2017, December 20). Rajneeshee bioterror attack . Homeland Security Digital Library. https://www.hsdl.org/c/timeline/rajn...terror-attack/
Pettus, E. W. (2022, September 2). EXPLAINER: Mississippi capital’s water woes are extensive. AP News . https://apnews.com/article/biden-mis...ea48a10c474df8
Prata, J. C., da Costa, J. P., Lopes, I., Duarte, A. C., & Rocha-Santos, T. (2020). Environmental exposure to microplastics: An overview on possible human health effects. Science of The Total Environment , 702 , 134455. https://doi.org/10.1016/j.scitotenv.2019.134455
Ragusa, A., Svelato, A., Santacroce, C., Catalano, P., Notarstefano, V., Carnevali, O., Papa, F., Rongioletti, M. C. A., Baiocco, F., Draghi, S., D’Amore, E., Rinaldo, D., Matta, M., & Giorgini, E. (2021). Plasticenta: First evidence of microplastics in human placenta. Environment International , 146 , 106274. https://doi.org/10.1016/j.envint.2020.106274
Ritchie, H., & Roser, M. (2020). CO₂ emissions. Our World in Data .
Ritchie, H., Samborska, V., & Roser, M. (2023). Plastic pollution. Our World in Data .
Runwal, P. (2024, May). The long-lasting legacy of Flint’s water crisis. American Chemical Society . https://cen.acs.org/environment/wate...uences/102/i14
Sadh, P. K., Duhan, S., & Duhan, J. S. (2018). Agro-industrial wastes and their utilization using solid state fermentation: A review. Bioresources and Bioprocessing , 5 (1). https://doi.org/10.1186/s40643-017-0187-z
Seabert, D., McKenzie, J. F., & Pinger, R. R. (2021). McKenzie’s An Introduction to Community & Public Health . Jones & Bartlett Learning.
Sherman, R. (2020). Large-Scale organic materials composting . NC State Extension Publications; AG-593, N.C. Cooperative Extension Service. https://content.ces.ncsu.edu/large-s...als-composting
Sierra Nevada Conservancy. (2022, January 24). 2021: Another historic Sierra Nevada fire season . Sierra Nevada Conservancy. https://sierranevada.ca.gov/2021-ano...a-fire-season/
SPLC. (2023, June 28). Mississippi city’s water problems stem from generations of neglect . Southern Poverty Law Center. https://www.splcenter.org/news/2023/...water-problems
Tailings . (2017, December 12). Earthworks. https://earthworks.org/issues/tailings/
The Associated Press. (2023, August 1). Phoenix ends 31-day streak of highs at or above 110 degrees — by reaching 108. NPR . https://www.npr.org/2023/08/01/11912...y-reaching-108
UNFCCC. (2015). ADOPTION OF THE PARIS AGREEMENT - Paris Agreement text English. United Nations .
USDA. (n.d.). Food waste faqs . USDA. Retrieved July 2, 2024, from https://www.usda.gov/foodwaste/faqs
USDA. (2018). Our history . Food Safety and Inspection Service; US Department of Agriculture. https://www.fsis.usda.gov/about-fsis/history
USGCRP. (2023a). Chapter 14: Air Quality. Fifth national climate assessment . U.S. Global Change Research Program. http://dx.doi.org/10.7930/nca5.2023.ch15
USGCRP. (2023b). Chapter 15 : Human health. Fifth national climate assessment . U.S. Global Change Research Program. http://dx.doi.org/10.7930/nca5.2023.ch15
USGCRP. (2023c). Chapter 22: Southeast . Fifth National Climate Assessment. https://nca2023.globalchange.gov/chapter/22/
(USGCRP) Hayden, M.H., P.J. Schramm, C.B. Beard, J.E. Bell, A.S. Bernstein, A. Bieniek-Tobasco, N. Cooley, M. Diuk-Wasser, Michael K. Dorsey, K.L. Ebi, K.C. Ernst, M.E. Gorris, P.D. Howe, A.S. Khan, C. Lefthand-Begay, J. Maldonado, S. Saha, F. Shafiei, A. Vaidyanathan, and O.V. Wilhelmi. (2023): Ch. 15. Human health. In: Fifth National Climate Assessment. Crimmins, A.R., C.W. Avery, D.R. Easterling, K.E. Kunkel, B.C. Stewart, and T.K. Maycock, Eds. U.S. Global Change Research Program, Washington, DC, USA. https://doi.org/10.7930/NCA5.2023.CH15
White, E. (2021, November 10). Judge approves $626 million settlement for Flint residents . PBS News. https://www.pbs.org/newshour/nation/...ter-litigation
White, E. (2022, December 9). Judge dismisses Flint water crisis charges against former Michigan governor . PBS News. https://www.pbs.org/newshour/politic...higan-governor
WHO. (n.d.). Estimating environmental health impacts . World Health Organization. Retrieved June 19, 2024, from https://www.who.int/activities/envir...health-impacts
WHO. (2020, July 2). Environmental health. World Health Organization . https://www.who.int/health-topics/en...alth#tab=tab_1
WHO. (2022, May 9). One health. World Health Organization: WHO . https://www.who.int/health-topics/on...alth#tab=tab_1
WHO. (2023a). Burden of disease attributable to unsafe drinking-water, sanitation and hygiene, 2019 update . Geneva: World Health Organization (WHO).
WHO. (2023b, December 15). Household air pollution. World Health Organization: WHO . https://www.who.int/news-room/fact-s...ion-and-health
Wiggins, A., & Morrow, M. (2022, June 14). ‘The Fearless 11’: The impact of white flight and integration in Jackson, Miss. Mississippi Free Press . https://www.mississippifreepress.org...-jackson-miss/
World Health Organization: WHO. (2019, July 30). Air pollution. World Health Organization: WHO . https://www.who.int/health-topics/ai...tion#tab=tab_2
Zachariah, M., Phillip, S., Pinto, I., Vahlberg, M., Singh, R., & Otto, F. E. L. (2023). Extreme heat in North America, Europe and China in July 2023 made much more likely by climate change . Imperial College London. https://doi.org/10.25561/105549
Zahran, S., McElmurry, S. P., Kilgore, P. E., Mushinski, D., Press, J., Love, N. G., Sadler, R. C., & Swanson, M. S. (2018). Assessment of the Legionnaires’ disease outbreak in Flint, Michigan. Proceedings of the National Academy of Sciences , 115 (8). https://doi.org/10.1073/pnas.1718679115
Zhang, X., Zhou, T., Zhang, W., Ren, L., Jiang, J., Hu, S., Zuo, M., Zhang, L., & Man, W. (2023). Increased impact of heat domes on 2021-like heat extremes in North America under global warming. Nature Communications , 14 (1). https://doi.org/10.1038/s41467-023-37309-y


18: Haruki Murakami (1949- )
Thumbnail: Haruki Murakami. (CC BY-SA 4.0; Galoren.com via Wikipedia )
selected template will load here
This action is not available.
Murakami Haruki (1949- ) The Second Bakery Attack Japanese Contemporary Literature "The Second Bakery Attack" is a short story by Murakami Haruki (surname: Murakami), a critically acclaimed and popular Japanese writer. This short story, first published in English in Playboy magazine in 1985, is also part of The Elephant Vanishes (1993), a collection of Murakami's seventeen short stories in English translation. "The Second Bakery Attack" displays many of Murakami's signature styles and approaches, such as mixing realism and surrealism (or magical realism), the use of interior monologue, allusions to popular culture and Western culture, and postmodernist elements. Along with the metaphor of the insatiable hunger the couple feels, what is noteworthy in the story is the newly married couple's changing gender dynamics upon the (partial) release of the husband's "secret," and the impact on the husband's psychology of these gender dynamics. Murakami's other notable works include Men without Women (2014), IQ84 (2009-10), Kafka on the Shore (2002), The Wind-up Bird Chronicle (1994-95), Norwegian Wood (1987), and A Wild Sheep Chase (1982). Many of his works have been translated into English (and their publication years are different from the original publication years listed above) by such translators as Philip Gabriel, Jay Rubin, and Alfred Birnbaum. Murakami is also a translator himself, having translated Western literary works into Japanese. Consider while reading:
Written by Kyounghye Kwon
Thumbnail: Haruki Murakami. (CC BY-SA 4.0; Galoren.com via Wikipedia )


33.1: Deriving Planck's Distribution Law
Albert Einstein developed a simple but effective analysis of induced emission and absorption of radiation along with spontaneous emission that can be used to derive the Planck formula for thermal radiation.
Consider two energy levels for the molecules in a material. The lower of the two is denoted as \(E_1\) and the higher as \(E_2\). The probability of a transition from level 1 up to level 2 through induced absorption is assumed to be proportional to the energy density per unit frequency interval, (\(du/d \nu\)). Likewise the probability of an induced transition from level 2 down to level 1 is assumed also to be proportional to (\(du/d\nu\)). These two probabilities are taken to be \(B_{12}(du/d\nu)\) and \(B_{21}(du/d\nu)\), respectively, where \(B_{12}\) and \(B_{21}\) are constants. The probability of a spontaneous emission is assumed to be a constant \(A_{21}\).
Let \(N_1\) and \(N_2\) be the number of molecules in energy states 1 and 2, respectively. For equilibrium the number of transitions from 1 to 2 has to be equal to the number from 2 to 1; i.e.,
\[\underbrace{N_1\left[B_{12}\left(\dfrac{du}{d\nu}\right)\right]}_{\text{flow up}} = \underbrace{N_2\left[B_{21}\left(\dfrac{du}{d\nu} \right)+ A_{21} \right]}_{\text{flow down}} \nonumber \]
This means that the ratio of the occupancies of the energy levels must be
\[\dfrac{N_2}{N_1} = \dfrac{B_{12}\left(\dfrac{du}{d\nu}\right)}{ B_{21}\left(\dfrac{du}{d\nu}\right) + A_{21}} \label{einstein2} \]
But the occupancies are given by the Boltzmann distribution as
\[N_1 = N_0 \exp \left(− \dfrac{E_1}{kT} \right) \nonumber \]
and
\[N_2 = N_0 \exp \left(−\dfrac{E_2}{kT} \right) \nonumber \]
where \(k\) is Boltzmann's constant and \(T\) is absolute temperature. \(N_0\) is just a constant that is irrelevant for the rest of the analysis.
Thus according to the Boltzmann distribution
\[\dfrac{N_2}{N_1} = \exp \left(−\dfrac{E_2−E_1}{kT} \right) \label{boltz2} \]
Therefore for radiative equilibrium, Equations \ref{boltz2} and \ref{einstein2} can be set to each other and
\[\exp\left(−\dfrac{E_2−E_1}{kT} \right) = \dfrac{B_{12}\left(\dfrac{du}{d\nu}\right)}{ B_{21}\left(\dfrac{du}{d\nu}\right) + A_{21}} \nonumber \]
This condition can be solved for \((du/dν)\); i.e.,
\[\dfrac{du}{d\nu} = \dfrac{A_{21}}{B_{12}\exp \left( \dfrac{E_2−E_1}{kT} \right)−B_{21}} \nonumber \]
Consider what happens to the above expression for as \(T \rightarrow \infty\). It goes to
\[\lim _ {T \rightarrow \infty} \dfrac{du}{d\nu} = \dfrac{A_{21}}{B_{12}−B_{21}} \nonumber \]
Einstein maintained that \((du/dν)\) must go to infinity as \(T\) goes to infinity. This requires that \(B_{12}\) be equal to \(B_{21}\).
Thus
\[\dfrac{du}{d\nu} = \dfrac{A_{21}/B_{21}}{\exp \left(\dfrac{E_2−E_1}{kT}\right)−1} \label{eq10} \]
Now Planck's assumption is introduced:
\[E_2−E_1 = hν \nonumber \]
Thus Equation \ref{eq10} becomes
\[\dfrac{du}{d\nu} = \dfrac{A_{21}/B_{21}}{\exp \left(\dfrac{hv}{kT}\right)−1} \label{eq11} \]
The Rayleigh-Jeans Radiation Law says
\[\dfrac{du}{d\nu}= \dfrac{8πkTν^2}{c^3} \label{RJ} \]
The Planck formula must coincide with the Rayleigh-Jeans Law for sufficiently small \(ν\). Note that the exponent in the denominator of Equation \ref{eq11} can be expanded (via a Taylor expansion ):
\[\exp\left(\dfrac{hν}{kT}\right) \approx 1 + \dfrac{hν}{kT} \nonumber \]
for sufficiently small \(ν\).
This means that Equation \ref{eq11} simplifies to
\[\dfrac{du}{d\nu} = \dfrac{A_{21}/B_{21}}{1 + (hν/kT) −1} = \dfrac{A_{21}/B_{21}}{hν/kT} \nonumber \]
and hence
\[\dfrac{du}{d\nu}= \left(\dfrac{A_{21}}{B_{21}} \right) \left( \dfrac{kT}{hν}\right) \label{eq20} \]
Equating Equations \ref{RJ} and \ref{eq20} for \((du/dν)\) gives
\[ \left(\dfrac{A_{21}}{B_{21}}\right) \left(\dfrac{kT}{hν}\right) = \dfrac{8πkTν^2}{c^3} \nonumber \]
which reduces to
\[\dfrac{A_{21}}{B_{21}} = \dfrac{8πhν^3}{c^3} \nonumber \]
Thus
\[\dfrac{du}{d\nu} = \dfrac{8πhν^³}{c^³} \dfrac{1}{\exp(hν/kT)−1} \nonumber \]
This is Planck's formula in terms of frequency.
Reference
- K.D. Möller, Optics , University Science Books, Mill Valley, California, 1988.


14.0: Integer Powers of 10
Nonnegative Integer Powers of 10
The phrase nonnegative integers refers to the set containing 0, 1, 2, 3, … and so on. In the expression , 10 is called the base , and 5 is called the exponent , or power . The exponent 5 is telling us to multiply the base 10 by itself 5 times. So, . By definition, any number raised to the 0 power is 1. So, .
In the following table, there are several nonnegative integer powers of 10 that have been written as a product. Notice that higher exponents result in larger products. What do you notice about the number of zeros in the resulting product?
| Exponential Form | Product | Number of Zeros in Product |
|---|---|---|
That’s right! The number of zeros is the same as the power each time!
Negative Integer Powers of 10
The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 10 is . We use negative exponents to indicate a reciprocal. For example, . Similarly, any expression with a negative exponent can be written with a positive exponent by taking the reciprocal. Several negative powers of 10 have been simplified in the table that follows. What do you notice about the number of zeros in the denominator (bottom) of each fraction?
| Exponential Form | Equivalent Simplified Expression | Number of Zeros in Denominator |
|---|---|---|
That’s right! The number of zeros is the same as the positive version of the power each time.
In the following table, we will write the same powers of 10 as decimals. Count the number of decimal places to the right of the decimal point. What do you notice?
| Exponential Form | Equivalent Simplified Expression | Number of Decimal Places to Right of Decimal |
|---|---|---|
That’s right! The number of decimal places to the right of the decimal point is the same as the positive version of the power each time.
Multiplying Integers by Positive Powers of 10
Did you know that the distance from the sun to Earth is over 90 million miles? This value can be represented as 90,000,000, or we can write it as a product: , which is actually a more compact way of writing 90 million. Notice that the power of 7 reflects the number of zeros in 90 million. Several products of positive integers and powers of 10 are given in the table that follows. Notice that the number of zeros is the same as the exponent except in one case.
| Exponential Form | Product | Number of Zeros in Product |
|---|---|---|
The only case in which the number of zeros didn’t equal the exponent was the last case. Why do you think that happened? That’s right! We multiplied by 70 which also had a zero. So, the product had a zero from the 70 and 5 zeros from for a total of 6 zeros in 7,000,000.
Multiplying by Negative Powers of 10
As we have seen, negative powers of 10 are decimals. Several products of positive integers and powers of 10 are given in the table below. Notice that multiplying an integer by 10 raised to a negative integer power results in a smaller number than you started with. Also, the number of decimal places to the right of the decimal point is the same as the exponent except in one case.
| Exponential Form | Product | Number of Decimal Places to Right of Decimal |
|---|---|---|
The only case in which the number of decimal places to the right of the decimal point didn’t equal the positive version of the exponent was the last case. Why do you think that happened? That’s right! We multiplied by 70, which ended in zero.
Moving the Decimal Place
A helpful shortcut when multiplying a number by a power of 10 is to “move the decimal point.” The following table shows several powers of 10, both positive and negative. Compare the location of the decimal point in the original number to the location of the decimal point in the product. How has it changed?
| Exponential Form | Product | How the Position of the Decimal Point Changed |
|---|---|---|
Notice that multiplying by a positive power of 10 moves the decimal point to the right, making the value larger, while multiplying by a negative power of 10 moves the decimal point to the left, making the value smaller. Also, the number of decimal places that the decimal point moves is exactly the positive version of the exponent.


2.3: Time Management in College
-
- Last updated
- Save as PDF
Questions to consider:
- Is time management different in college from what I am used to?
- How different is college schoolwork from high school work?
You may find that time management in college is very different from anything you have experienced previously. For the last 12 years, almost all your school time was managed by educators and your parents. What you did and when you did it was controlled by others. In many cases, even after-school time was set by scheduled activities (such as athletics) and by nightly homework that was due the next day.
In the workplace, the situation is not very different, with activities and time on task being monitored by the company and its management. This is so much a part of the working environment that many companies research how much time each task should take, and they hold employees accountable for the time spent on these job functions. In fact, having these skills will help you stand out on the job and in job interviews.
| K–12 | College |
| Many class activities are planned. | Class time is given to receiving information. |
| Homework is often similar for each student. | You may have freedom in homework choices. |
| Time is managed by others more often. | Time is managed by the student. |
In college, there is a significant difference because a great deal of time management is left up to you. While it is true that there are assignment due dates and organized classroom activities, learning at the college level requires more than just the simple completion of work. It involves decision-making and the ability to evaluate information. This is best accomplished when you are an active partner in your own learning activities.
As an example of how this works, think about a college assignment that involves giving a classroom presentation. To complete the assignment, you are given time to research and reflect on the information found. As a part of the assignment, you must reach your own conclusions and determine which information that you have found is best suited for the presentation. While the date of the actual presentation and how long it will last are usually determined by the instructor, how much time you spend gathering information, the sources you use, and how you use them are left to you.
WHAT STUDENTS SAY
-
How difficult is it for you to keep track of multiple tasks over the course of a term?
- Extremely easy
- Somewhat easy
- Somewhat difficult
- Extremely difficult
-
Do you use a particular app to help you manage your time?
- I use Google calendar
- I use the calendar on my phone
- I use a paper/notebook planner
- I use the calendar on my learning management system
- I use another app or system
- I don't use any type of planner or app
-
Rank the following in terms of what you would most like to improve regarding your time management skills.
- My ability to predict how much time my tasks will take.
- My ability to balance various obligations.
- My ability to avoid procrastination.
- My ability to limit distractions.
You can also take the anonymous What Students Say surveys to add your voice to this textbook. Your responses will be included in updates.
Students offered their views on these questions, and the results are displayed in the graphs below.
How difficult is it for you to keep track of multiple tasks over the course of a term?
Do you use a particular app to help you manage your time?
Rank the following in terms of what you would most like to improve regarding your time management skills.
You Have Lots of Time to Manage
For college-level learning, this approach is important enough that you can expect to spend much more time on learning activities outside the classroom than you will in the classroom. In fact, the estimated time you should spend will be at least two hours of outside learning for every one hour of lecture. Some weeks may be more intense, depending on the time of the semester and the courses you are taking. If those hours are multiplied over several courses in a given session, you can see how there is a significant amount of time to manage. Unfortunately, many students do not always take this into consideration, and they spend far less time than is needed to be successful. The results of poor time management are often a shock to them.
“In college, as an active participant in your own education, what you do and when you do it is largely determined by you.”
The Nature of What You Have to Do Has Changed
Returning to our example of the classroom-presentation assignment, you can see that the types of learning activities in college can be very different from what you have experienced previously. While there may have been similar assignments in high school, such as presentations or written papers, the level of expectation with length and depth is significantly different in college. This point is made very clear when comparing facts about the requirements of high school work to the type of work students produce in college. One very strong statistic that underscores this comes from a study conducted by the Pew Research Center. They found that 82 percent of teens report that their typical high school writing assignments were only a single paragraph to one page in length. 2 (Writing Technology and Teens, 2004, Pew Research Center) This is in stark contrast to a number of sources that say that writing assignments in lower-level college courses are usually 5–7 pages in length, while writing assignments in upper-level courses increase to 15–20 pages.
It is also interesting to note that the amount of writing done by a college student can differ depending on their program of study. The table below indicates the estimated average amount of writing assigned in several disciplines. To estimate the number of pages of assigned writing, the average number of writing assignments of a given page length was multiplied by an approximate number of pages for the assignment type (see Estimating Number of Pages Written for calculation details).
| Writing Assignments Vary in Length | |
|---|---|
| Discipline | Number of Pages Assigned in Introductory Course |
| Arts & Humanities | 49 |
| Biological Sciences, Agriculture, & Natural Resources | 47 |
| Physical Sciences, Mathematics, & Computer Science | 44 |
| Social Sciences | 52 |
| Business | 48 |
| Communications, Media, & Public Relations | 50 |
| Education | 46 |
| Engineering | 46 |
| Health Professions | 43 |
| Social Service Professions | 47 |
High school homework often consists of worksheets or tasks based on reading or classroom activities. In other words, all the students are doing the same tasks, at relatively the same time, with little autonomy over their own education.
Using the earlier example of the presentation assignment, not only will what you do be larger in scale, but the depth of understanding and knowledge you will put into it will be significantly more than you may have encountered in previous assignments. This is because there are greater expectations required of college graduates in the workplace. Nearly any profession that requires a college degree has with it a level of responsibility that demands higher-level thinking and therefore higher learning. An often-cited example of this is the healthcare professional. The learning requirements for that profession are strict because we depend on those graduates for our health and, in some cases, our lives. While not every profession may require the same level of study needed for healthcare, most do require that colleges maintain a certain level of academic rigor to produce graduates who are competent in their fields.
Footnotes
- 2 http://www.pewinternet.org/2008/04/2...ogy-and-teens/
- 3 nsse.indiana.edu/html/sample_...of_writing.cfm


13.1.2: The Telescope
Looking Up!
As technology developed, so did ways to observe and model what people saw in the heavens. Observing included ways to find objects — star maps and globes — as well as ways to track objects, like the astrolabe. The invention of the telescope, and astronomical use by Galileo in 1609, opened up the heavens in a new, exciting way. And, as technology developed, ways to present these astronomical objects became more and more exciting.
The Telescope
Initially, only those who were well to do or royalty had access to telescopes. And, astronomers were often hired by royalty or rewarded for their discoveries. Yet, as time went on, and methods for producing telescopes became more common, many universities and cities across Europe and the United States in particular saw an observatory with a refracting telescope as a cornerstone. The main challenge was producing excellent — or even good — optics. By the 18 th Century, refracting telescopes were being produced in some number by opticians like John Dollond. In the 19 th century, opticians such as Henry Fitz and Alvan Clark produced fine commercial telescopes. Alvan Clark & Sons made what is still the largest refracting telescope in the world, the 40-inch refractor at Yerkes Observatory, Williams Bay, WI.
Reflecting telescope were not initially popular, due to the difficulties producing the right type of curve on the primary mirror, and keeping the surface reflective. With the development of techniques to deposit a thin coating of aluminum on a surface of glass, the reflector not only became commercially viable, but less expensive to manufacture than the refractor.